Determine whether f(x)=- 2 + x is increasing or decreasing on (- … - Vidyadip

Question

Determine whether $f(x)=-\frac{\pi}{2}+ \sin x$ is increasing or decreasing on $\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$

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Answer

$f(x)=-\frac{x}{2}+\sin x$
$\Rightarrow f^{\prime}(x)=-\frac{1}{2}+\cos x$
Now
$x \in\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$
$\Rightarrow-\frac{\pi}{3} < x < \frac{\pi}{3}$
$\Rightarrow \cos \left(-\frac{\pi}{3}\right) < \cos x < \cos \frac{\pi}{3}$
$\Rightarrow \cos \left(\frac{\pi}{3}\right) < \cos x < \cos \frac{\pi}{3}$
$\Rightarrow \frac{1}{2} < \cos x < \frac{1}{2}$
$\Rightarrow-\frac{1}{2}+\cos x > 0$
$\Rightarrow f^{\prime}(x) > 0$
Hence, $f(x)$ is an increasing function on $\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$.

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